Existence and Uniqueness of Normalized Multi-peak Solutions for Coupled Nonlinear Schr\"odinger Systems

We consider the following two-component coupled nonlinear Schr\"odinger (CNLS) system: $$\begin{cases} -\Delta u +(P(x) + \lambda ) u=\mu_1 u^3+\beta u v^2, & \text{in } \mathbb{R}^N,\\ -\Delta v +(Q(x) + \lambda ) v =\mu_2 v^3+\beta vu^2, & \text{in } \mathbb{R}^N \end{cases}$$ with the mass constraint $\int_{\mathbb{R}^N} (u^2+v^2)\,dx = \rho^2$ for $N=2,3$, where $\rho>0$ is a parameter. By employing the Lyapunov-Schmidt reduction and local Pohozaev identities, we establish the existence and local uniqueness of normalized multi-peak solutions: the result holds for sufficiently small $\rho$ when $N=3$, and for $\rho$ approaching a critical threshold when $N=2$. The main difficulty lies in that the mass constraint involves interactions among all concentration points, while a more refined characterization of such normalized solutions further requires sharp order estimates. In this work, we have discovered some new phenomena that differ from those of solutions without mass constraint and single-peak solutions.